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\begin{document}
\title{Discrete Diagonal Propagation and Isotropy\\\large Physicist Reproduction Note}
\author{Frostyverse Laboratory}
\date{Website Laboratory v1.0.48}
\maketitle

\statusbox{\textbf{Scientific status.} Historical software candidate / software-sufficiency test. This note documents the exact free-FQ/vector-conserving-Shore calculation exposed by the public Python Test ZIP. A PASS means the declared candidate survived the stated numerical attacks. It does not establish that nature contains FrostNodes, FrostCells, Shore, or a discrete FrostGrid. Later Frostyverse transport interpretations must not be silently back-ported into this preserved historical simulator.}

\begin{abstract}
This Laboratory item attacks the original cubic-grid diagonal objection in a directly executable form. A continuous carried direction is supplied to a discrete FrostCell handoff rule whose completed outcomes can occur only at legal FrostNodes. The candidate constructs a probability distribution over those discrete outcomes such that the exact ensemble mean preserves the carried FrostVector, then asks whether finite completed hops develop persistent axis/diagonal bias, radial distortion, or FrostCell-boundary pathology. The public Python wrapper imports the unchanged historical Reference Simulator engine and exposes arbitrary $X/Y/Z$ directions, exact boundary branches, custom sample counts, and many-direction spherical stresses.
\end{abstract}

\section{What is being tested---and what is not}
Let the carried nonzero FrostVector be
\[
\mathbf v=(v_x,v_y,v_z),
\qquad
\vhat=\frac{\mathbf v}{\lVert\mathbf v\rVert}.
\]
The candidate does \emph{not} require a single discrete hop to point exactly along $\vhat$. Instead, a completed handoff terminates at one legal corner of one compatible signed FrostCell, while the probability-weighted ensemble of such completions must preserve $\vhat$.

This distinction matters. The exact ensemble-mean agreement is deliberately constrained by the Shore rule and therefore is \textbf{not by itself evidence for isotropy}. The useful attack is whether finite discrete completions show systematic direction dependence after that constraint is imposed.

This Lab also has no GR/SR target equation. Its scoring quantities are the prescribed carried direction, finite-sample angular residuals, radial/magnitude residuals, and branch consistency. The separate Rotational Invariance Lab performs a stronger whole-setup rotation audit.

\section{Discrete FrostCell geometry}
For each Cartesian component of $\vhat$, the sign selects the corresponding neighboring cell. If a component is numerically zero, both adjacent signs are legal. With signed cell label
\[
\mathbf s=(s_x,s_y,s_z),\qquad s_i\in\{-1,+1\},
\]
the seven nonzero completed-node offsets are
\[
\mathcal N(\mathbf s)=
\left\{(s_xb_x,s_yb_y,s_zb_z):
 b_i\in\{0,1\},\ (b_x,b_y,b_z)\neq(0,0,0)
\right\}.
\]
Thus an interior direction has one compatible FrostCell; an exact face-boundary direction has two; and an exact coordinate-axis direction has four. The public custom wrapper evaluates \emph{every} compatible branch rather than silently selecting one.

All geometry in this test is normalized to a unit FrostCell edge. It is a dimensionless software model; no physical FrostGrid length scale is inferred here.

\section{The actual Frostyverse Shore calculation}
The following equations summarize the calculation implemented in the unchanged historical simulator source, \path{frostyverse_simulator.py}, and called by the public wrapper.

\subsection{Temporary directed smear influence}
For candidate offset $\mathbf r_n\in\mathcal N$, define the axial and transverse coordinates relative to $\vhat$:
\[
s_n=\mathbf r_n\cdot\vhat,
\qquad
\boldsymbol\rho_n=\mathbf r_n-s_n\vhat,
\qquad
\rho_n=\lVert\boldsymbol\rho_n\rVert.
\]
If $s_n\le0$, the directed influence is zero. Otherwise, during normalized within-update integration time $\tau\in(0,1]$, the historical support law uses
\[
h(\tau)=h_{\rm reach}\,\tau^{p_h},
\qquad
b_n(\tau)=\max\!\left(0,h(\tau)-s_n\right),
\]
with a tapered transverse radius
\[
R_n(\tau)=R_{\rm tail}
+(R_{\rm head}-R_{\rm tail})
\exp\!\left[-\frac{b_n(\tau)}{L_{\rm taper}}\right].
\]
The instantaneous candidate influence is
\[
I_n(\tau)=
\exp\!\left[-\frac12\left(\frac{\rho_n}{R_n(\tau)}\right)^2\right]
\exp\!\left[-\frac12\left(\frac{s_n-h(\tau)}{\sigma_\parallel}\right)^2\right]
\left(1+g_{\rm vac}\tau\right).
\]
The public reference uses the frozen historical values
\[
\begin{aligned}
h_{\rm reach}&=\sqrt3, & p_h&=1.70,\\
R_{\rm head}&=0.55, & R_{\rm tail}&=0.18,\\
L_{\rm taper}&=0.60, & \sigma_\parallel&=0.25,\\
g_{\rm vac}&=1.00, & p_{\rm claim}&=1.00.
\end{aligned}
\]
These are assumptions of this historical software candidate, not measured physical constants.

Using $M$ integration substeps, the full-update influence is the Riemann sum
\[
A_n=\frac1M\sum_{k=1}^{M}I_n(k/M),
\]
and the raw normalized Shore claim is
\[
q_n=\frac{A_n^{p_{\rm claim}}}
{\sum_{m\in\mathcal N}A_m^{p_{\rm claim}}}.
\]
The default public reproduction uses $M=96$.

\subsection{Vector-conserving Shore projection}
The raw claims $q_n$ need not preserve the carried vector exactly. The final Shore distribution $p_n$ is the nonnegative probability vector closest to $q_n$ in Euclidean probability space while satisfying
\[
\boxed{
\sum_{n\in\mathcal N}p_n=1,
\qquad p_n\ge0,
\qquad
\sum_{n\in\mathcal N}p_n\mathbf r_n=\vhat.
}
\]
Operationally,
\[
\boxed{
p=\underset{p}{\arg\min}\;
\sum_{n\in\mathcal N}(p_n-q_n)^2}
\]
subject to those constraints. The historical implementation enumerates admissible supports and rejects any support that requires a negative probability. The reported Shore correction is
\[
L_{2,\rm Shore}
=
\left[\sum_n(p_n-q_n)^2\right]^{1/2}.
\]
Consequently,
\[
\mathbb E[\mathbf r]
=
\sum_n p_n\mathbf r_n
=\vhat
\]
to solver tolerance. Again, this exact equality is a property of the declared candidate rule, not a blind prediction.

\section{Finite completed-hop observable}
The actual numerical isotropy attack samples $N$ completed endpoints from the frozen distribution $p_n$. If the sampled offsets are $\mathbf r_{k}$,
\[
\bar{\mathbf d}_N
=\frac1N\sum_{k=1}^{N}\mathbf r_k.
\]
The primary finite-sample angular residual is
\[
\boxed{
\theta_N
=
\cos^{-1}\!\left(
\frac{\bar{\mathbf d}_N\cdot\vhat}
{\lVert\bar{\mathbf d}_N\rVert}
\right)
}
\]
and the radial/magnitude diagnostic is
\[
r_N=\lVert\bar{\mathbf d}_N\rVert.
\]
If the candidate has a hidden axis or diagonal preference, finite-sample residuals should reveal orientation-dependent structure as directions, seeds, and sample counts are varied.

\section{Published focused reference calculation}
The public Python Test reproduces the frozen focused suite with
\begin{center}
\begin{tabular}{ll}
\toprule
Quantity & Public reference value \\
\midrule
Base carried direction & $(1,0.37,0.12)$, normalized \\
Rotation axis for direction sweep & $(0.31,0.77,0.55)$, normalized \\
Direction-sweep orientations & 16 \\
Completed hops / swept direction & 5000 \\
Fibonacci-sphere directions & 64 \\
Completed hops / sphere direction & 2500 \\
Base seed & 7 \\
Smear integration substeps & 96 \\
\bottomrule
\end{tabular}
\end{center}

The 16 direction-sweep vectors are generated by Rodrigues rotation of the base vector about the stated axis. The 64 broader directions are generated by a rotated Fibonacci-sphere construction. These are sampling devices; they are not new physics inputs.

Fresh execution of the distributed Python ZIP reproduces
\begin{center}
\begin{tabular}{lr}
\toprule
Observable & Result \\
\midrule
Direction-sweep angular mean / max & $0.35148^\circ / 0.85120^\circ$ \\
Sphere angular mean / max & $0.54079^\circ / 1.48100^\circ$ \\
Sphere magnitude mean / std. dev. & $1.000545 / 0.006337$ \\
Verdict & PASS \\
\bottomrule
\end{tabular}
\end{center}
The deterministic reference summary seals to
\begin{quote}\footnotesize
\path{9d4f3623d061d9caee0af91b7f5cc22c9ae733c01df2725f1e09a3c225eaf9e4}.
\end{quote}

\section{Predeclared software gates}
The focused reference uses
\begin{align*}
\max(\theta_{\rm direction\ sweep}) &< 2.0^\circ,\\
\max(\theta_{\rm sphere}) &< 2.5^\circ,\\
\left|\operatorname{mean}(r_{\rm sphere})-1\right| &< 0.02.
\end{align*}
These thresholds score the finite sampled outputs. They are not supplied to the Shore projection or used to construct $p_n$.

\section{Fresh custom-input attacks}
The public wrapper deliberately exposes the quantities a reviewer is most likely to challenge.

\subsection{Arbitrary off-axis direction}
For example,
\begin{verbatim}
python lab_test.py --direction 1 0.271 0.619 --ticks 10000 --seed 404
\end{verbatim}
normalizes the requested direction to approximately
\[
(0.828570657,\ 0.224542648,\ 0.512885237).
\]
A fresh run produced one compatible FrostCell branch, exact Shore-mean component error $2.78\times10^{-17}$, finite-hop angular error $0.479232^\circ$, and sample magnitude $0.99979098$.

\subsection{Exact FrostCell-boundary branch attack}
An exact axis case is more revealing than an ordinary interior vector:
\begin{verbatim}
python lab_test.py --direction 1 0 0 --ticks 10000 --seed 404
\end{verbatim}
The wrapper reports all four compatible cells
\[
(+,-,-),\quad (+,-,+),\quad (+,+,-),\quad (+,+,+).
\]
In the tested run, every branch returned the exact axis mean with $0^\circ$ finite-hop angular error and unit sample magnitude. A reviewer should change the axis/face case, seed, and sample count rather than treating this single example as proof.

\subsection{Fresh many-direction stress}
A broader stress can be requested directly:
\begin{verbatim}
python lab_test.py --sphere-stress 96 --ticks 5000 --seed 9001
\end{verbatim}
A fresh run gave
\begin{align*}
\text{angular mean / max} &= 0.409335^\circ / 1.162597^\circ,\\
\text{radius mean / std. dev.} &= 0.99975148 / 0.00511363,
\end{align*}
within the published spherical gates.

\section{What would break the claim}
A reviewer should reopen or reject this candidate if a reproducible attack shows, for example:
\begin{itemize}
\item persistent finite-hop residuals correlated with grid axes, face diagonals, or body diagonals;
\item increasing rather than shrinking angular/radial bias as completed-hop count is raised;
\item incompatible behavior among mathematically legal boundary branches;
\item a direction family for which no nonnegative vector-conserving Shore mixture exists under the frozen rules;
\item sensitivity that appears only because one legal branch was hidden or silently discarded;
\item dependence on an external smooth-space target that is not already the declared carried FrostVector input.
\end{itemize}

\section{Reproduction commands}
Python 3.10 or newer is recommended. The package uses only the Python standard library; no third-party dependency is required.

Interactive launcher:
\begin{verbatim}
python lab_test.py
\end{verbatim}
Published focused reference:
\begin{verbatim}
python lab_test.py --reference
\end{verbatim}
Custom arbitrary direction:
\begin{verbatim}
python lab_test.py --direction X Y Z --ticks 10000 --seed 404
\end{verbatim}
Custom many-direction stress:
\begin{verbatim}
python lab_test.py --sphere-stress 96 --ticks 5000 --seed 9001
\end{verbatim}
Run the unchanged complete historical suite:
\begin{verbatim}
python lab_test.py --historical
\end{verbatim}
Verify the preserved historical archive:
\begin{verbatim}
python lab_test.py --verify
\end{verbatim}
Focused runs save human-readable TXT and machine-readable JSON records under \code{results/}.

\section{Package and provenance}
The public Python Test ZIP contains
\begin{itemize}
\item \code{lab\_test.py} --- public focused/custom wrapper;
\item \code{README\_FIRST.md} and \code{EXPECTED\_RESULTS.txt};
\item \code{run\_test.bat} and \code{run\_test.sh};
\item \code{requirements.txt} --- documents that no third-party libraries are required;
\item \code{SHA256SUMS.txt};
\item \code{historical/Frostyverse\_Reference\_Simulator\_v1.0.zip} --- unchanged historical engine/package.
\end{itemize}

Historical archive SHA-256:
\begin{quote}\footnotesize
\path{2c6bd32527c4fa71dbe1b55b4b00843dfd6d50b91e9227626e0b1b7991cdb807}
\end{quote}
The complete public Python Test ZIP used for this Laboratory has SHA-256
\begin{quote}\footnotesize
\path{7f150b7cc01f803bf59e780a0129cbbc00c8efdbbc3264cff8c93525b5c37e51}
\end{quote}
The public wrapper verifies the historical archive hash before importing the engine. It does not replace the mechanics with a simplified surrogate.

\section{Browser demonstration versus scientific reproducer}
The browser Laboratory is intentionally visual and useful for orientation. Its JavaScript sampling sequence need not match Python's pseudorandom sequence. The Python Test ZIP and unchanged historical Reference Simulator are the primary reproducibility artifacts for this Lab.

\section{Claim boundary}
The publication-safe conclusion is narrow:
\begin{quote}
The specified historical vector-conserving discrete-completion candidate survived the declared finite direction/isotropy tests, including arbitrary off-axis inputs, a broad spherical direction sample, and explicit legal FrostCell-boundary branches.
\end{quote}
It is \textbf{not} established here that a physical FrostGrid exists, that the Shore postulate follows from lower-level nature, that exact ensemble-mean preservation is independent evidence for isotropy, or that every continuum/high-energy regime is isotropic. The software is published so those stronger possibilities can be attacked rather than assumed.

\end{document}